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We continue to investigate correspondences between, on the one hand, scattering amplitudes for massive higher-spin particles and gravitons in appropriate quantum-to-classical limits, and on the other hand, classical gravitational interactions of spinning black holes according to general relativity. We first construct an ansatz for a gravitational Compton amplitude, at tree level, constrained only by locality, crossing symmetry, unitarity and consistency with the linearized-Kerr 3-point amplitude, to all orders in the black hole's spin. We then explore the extent to which a unique classical Compton amplitude can be identified by comparing with the results of the classical process of scattering long-wavelength gravitational waves off an exact Kerr black hole, determined by appropriate solutions of the Teukolsky equation. Up to fourth order in spin, we find complete agreement with a previously conjectured exponential form of the tree-level Compton amplitude. At higher orders, we extract tree-level contributions from the Teukolsky amplitude by an analytic continuation from a physical ($a/GM<1$) to a particle-like ($a/GM>1$) regime. Up to the sixth order in spin, we identify a unique \textit{conservative} part of the amplitude which is insensitive both to the choice of boundary conditions at the black hole horizon and to branch choices in the analytic continuation. The remainder of the amplitude is determined modulo an overall sign from a branch choice, with the sign flipping under exchanging purely ingoing and purely outgoing boundary conditions at the horizon. Along the way, we make contact with novel applications of massive spinor-helicity variables pertaining to their relation to EFT operators and (spinning) partial amplitudes.